Power of Complex Numbers
Calculation of \(z^w\) with complex base and complex exponent
Power Calculator
Power of Complex Numbers
The power \(z^w\) is calculated by \(z^w = e^{w \ln z}\). Both base and exponent can be complex.
Power - Properties
General Formula
Conversion to exponential function and logarithm
For real exponents
With polar form \(z = re^{i\phi}\)
Important Properties
- \(z^0 = 1\) (for \(z \neq 0\))
- \(z^1 = z\)
- \(z^{-1} = \frac{1}{z}\)
- \((z^{w_1})^{w_2} \neq z^{w_1 w_2}\) (in general!)
Multivaluedness
For complex exponents, \(z^w\) is multivalued due to the multivaluedness of the logarithm! This calculator returns the principal value.
Special Cases
- Integer exponents: \(z^n\) (unique)
- Rational exponents: \(z^{p/q}\) (q values)
- Real exponents: \(z^r\) (infinitely many values)
- Complex exponents: \(z^w\) (infinitely many values)
Formulas for Power of Complex Numbers
The power of a complex number with complex exponent is defined through the exponential function and the logarithm.
General Definition
Using the principal value of the complex logarithm
With Polar Form
For \(z = re^{i\phi}\) and \(w = c + di\)
Calculation Examples
Example 1: \((1+i)^2\) (real exponent)
Method 1: Expansion
\((1+i)^2 = (1+i)(1+i)\)
\(= 1 + i + i + i^2\)
\(= 1 + 2i - 1 = 2i\)
Method 2: Polar Form
\(1+i = \sqrt{2}e^{i\pi/4}\)
\((1+i)^2 = (\sqrt{2})^2 e^{i\cdot 2\pi/4}\)
\(= 2e^{i\pi/2} = 2i\) ✓
Example 2: \(i^i\) (complex exponent)
Calculation:
\(i = e^{i\pi/2}\)
\(\ln(i) = i\pi/2\)
\(i^i = e^{i \ln(i)} = e^{i \cdot i\pi/2}\)
\(= e^{-\pi/2}\)
\(\approx 0.208\) (real!)
Example 3: \(2^{1+i}\)
Calculation:
\(\ln(2) \approx 0.693\)
\(2^{1+i} = e^{(1+i)\ln 2}\)
\(= e^{\ln 2 + i\ln 2}\)
\(= e^{\ln 2} \cdot e^{i\ln 2}\)
\(= 2 \cdot (\cos(0.693) + i\sin(0.693))\)
\(\approx 1.54 + 1.28i\)
Example 4: \((1+i)^{1+i}\)
Calculation:
\(1+i = \sqrt{2}e^{i\pi/4}\)
\(\ln(1+i) = \ln\sqrt{2} + i\pi/4\)
\(\approx 0.347 + 0.785i\)
\((1+i)^{1+i} = e^{(1+i)(0.347+0.785i)}\)
\(= e^{(0.347-0.785) + i(0.785+0.347)}\)
\(\approx 0.274 + 0.584i\)
Special Cases and Computation Rules
Integer Exponents
For integer \(n\), \(z^n\) is unique:
\(z^2 = z \cdot z\)
\(z^3 = z \cdot z \cdot z\)
\(z^{-n} = \frac{1}{z^n}\)
With polar form:
\(z^n = r^n e^{in\phi}\)
Roots (rational exponents)
\(z^{1/n} = \sqrt[n]{z}\) has \(n\) different values:
\[z_k = \sqrt[n]{r} \cdot e^{i(\phi + 2\pi k)/n}\]
with \(k = 0, 1, ..., n-1\)
This calculator returns the principal value (k=0)
Caution with computation rules!
NOT always valid:
❌ \(z^{w_1 + w_2} = z^{w_1} \cdot z^{w_2}\)
❌ \((z^{w_1})^{w_2} = z^{w_1 w_2}\)
❌ \((z_1 z_2)^w = z_1^w \cdot z_2^w\)
Valid only for:
✅ Integer exponents
✅ Positive real bases
Multivaluedness
The complex logarithm is multivalued:
\[\ln z = \ln|z| + i(\arg z + 2\pi k)\]
with \(k \in \mathbb{Z}\)
Therefore: \(z^w = e^{w(\ln|z| + i(\arg z + 2\pi k))}\)
has infinitely many values for non-integer w!
Power of Complex Numbers - Detailed Description
Definition
The power of a complex number with complex exponent is defined via the exponential function and the logarithm:
1. Logarithm of base: \(\ln z\)
2. Multiplication with exponent: \(w \ln z\)
3. Apply exponential function: \(e^{w \ln z}\)
Calculation with Polar Form
In polar form, the calculation is often simpler:
Let \(z = re^{i\phi}\) and \(w = c + di\), then:
Magnitude: \(r^c e^{-d\phi}\)
Argument: \(c\phi + d\ln r\)
Practical Applications
Complex powers find applications in many fields:
• Fractals: Mandelbrot set, Julia sets
• Quantum mechanics: energy eigenvalues
• Signal processing: frequency analysis
• Differential equations: complex solutions
Visualization
The power function \(f(z) = z^w\) for fixed w:
- Integer w: \(w\)-fold rotation and scaling
- Rational w = p/q: q-sheeted Riemann surface
- Real w: spiral in the complex plane
- Complex w: complex deformation
Principal Value
Since the logarithm is multivalued, we define the
principal value of the power by using the
principal value of the logarithm:
\[\text{Log}(z) = \ln|z| + i\arg(z)\]
with \(-\pi < \arg(z) \leq \pi\)
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